This is an example of a rational domain in \(\mathbb{A}^1\). The importance of rational domains lies in the following theorem:

Theorem (Gerritzen-Grauert). Let \(U\) be an affinoid domain. Every affinoid subdomain of \(U\) is a finite union of rational subdomains of \(U\).

An affinoid algebra is a quotient of a Tate algebra by a closed ideal. Global analytic functions on affinoid domains are called affinoid functions. Recall that a function is called meromorphic on a domain if it can be locally represented as a quotient of analytic functions. We will see that on affinoid domains meromorphic functions are quotients of affinoid functions globally.

Intersections of affinoid domains are affinoid. Affinoids play the role of opens in rigid geometry. In fact, one can define the ``rigid \(G\)-topology’’: it is a Grothendieck topology where admissible covers are finite covers of \(\mathrm{Specm} A\) by affinoid domains.

Note that \(x\) and \(y\) cannot be extended to a function meromorphic on \(\mathbb{A}^1\) (or unit disc), since their poles converge to 0, and meromorphic functions can only have isolated poles (the proof is similar to the Archimedean case).

Quite expectedly though, they can be uniquely extended to the punctured unit disc. It took me some effort to find out how this can be proved: analytic continuation works quite differently in non-Archimedean situation than in the complex-analytic setting.

Proof. Let \(f_1, f_2\) be two meromorphic functions on \(A_2\) that coincide on \(A_1\), they are then quotients of pairs of affinoid on \(A_n\) functions, \(f_1 = \dfrac{g_1}{h_1}, f_2 = \dfrac{g_2}{h_2}\). Then \(g_1 h_2 = g_2 h_2\) on \(A_1\), but then by Tate acyclicity theorem \(g_1 h_2 = g_2 h_1\) on \(A_2\), and therefore \(\dfrac{g_1}{h_1} = \dfrac{g_2}{h_2}\).

In their 1987 paper van den Dries and Denef have introduced subanalytic domains and functions over the field of p-adic numbers, and have proved a quantifier elimination result for \(\mathbb{Q}_p\) equipped with restricted subanalytic functions. The language of the theory they define consists of the Denef-Pas language, extended with graphs of \(D\)-functions, defined inductively as follows.

the composition \(f(g_1(\ldots), \ldots, g_n(\ldots))\) of a \(D\)-function \(f(x_1, \ldots, x_n)\) with \(D\)-functions \(g_1, \ldots, g_n\) is a \(D\)-function.

The language of valued rings expanded with symbols of \(D\)-functions, naturally interpredet, is called \(L^D_{an}\), and Denef and van den Dries have shown that in this language, definable subsets of \({\mathbb{Z}}_p^n\) are quantifier free definable.

Here is an illustration of expressiveness of this expansion: restrictions of Weirstrass ellptic functions to their fundamental domain are definable.

Firstly, observe that any affinoid domain is definable. Indeed, by Gerritzen-Grauert theorem an affinoid domain is a union of rational domains, and these can be represented as projections of a vanishing set of a family of affinoid functions on a polydisc. Secondly, it immediately follows that affinoid functions are also definable (restrict functions on the polydisc to the closed subset).

Now Weirstrass elliptic functions are meromorphic on the annulus \(0 \leq v(x) \leq v(q)\), so they are quotients of an analytic — hence affinoid — functions by polynomials, therefore, they are definable in \(L^D_{an}\).